I have a doubt. Is it possible to get the cumulants of a probability distribution from the autocorrelation function? or the probability distribution?. For example, the variance (the second cumulant) corresponds to the autocorrelation function at $\tau =0$. Thank you so much
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Cumulants from the probability distribution: of course. The cumulants are defined in terms of the expectations of particular functions of the random variable.
An autocorrelation function is a property of a stochastic process, i.e., a family of different random variables over some time parameter. The autocorrelation function tells you the covariance (or correlation) of different random variables at different times. It's intended to capture something about the interrelatedness of those random variables. You're right, you can get the variance of any particular random variable in the stochastic process from the autocorrelation function, but you can't get any other moments (or cumulants). In terms of knowledge of the distributions of the stochastic process, the autocorrelation provides a tiny drop in the ocean that moments or cumulants or characteristic functions or generating functions can provide.
But for the case of Gaussian processes, the autocorrelation function (along with means) does indeed completely determine the stochastic process.